Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations with Wave Operator (Journal of Computational Mathematics, 2024)
Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It isan explicit and unconditionally von Neumann stable scheme. However, there has been noresearch work on numerical solutions of nonlinear Schrödinger equations with wave operatorby using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class ofinvariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator. Byusing the discrete energy method, it is shown that their solutions possess the discrete energyand mass conservative laws, and conditionally converge to exact solutions with an order of $\mathcal{O}(τ^2+h^2_x+(τ/h_x)^2)$ for 1D problem and an order of $\mathcal{O}(τ^2+h^2_x+h^2_y+(τ/h_x)^2+(τ/h_y)^2)$ for 2D problem in $H^1$-norm. Here, $τ$ denotes time-step size, while, $h_x$ and $h_y$ representspatial meshsizes in $x$- and $y$-directions, respectively. Then, by introducing a stabilizedterm, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses, but also own much better stabilitythan original schemes. Finally, numerical results demonstrate the theoretical analyses.
Keywords: Nonlinear Schrödinger equations with wave operator; Du Fort-Frankel finite difference methods; Discrete energy and mass conservative laws; Numerical convergence.